# Zero group refractive index

Tags:
1. May 10, 2016

### jsea-7

1. The problem statement, all variables and given/known data.
I am attempting to determine the group refractive index of a laser cavity at it's resonance frequency.

2. Relevant equations.
\begin{align*}
\frac{2\omega n L}{c} &= 2m\pi
\end{align*}

\begin{align*}
n_g &= n + \omega \frac{dn}{d\omega}
\end{align*}

3. The attempt at the solution.
I have considered a uniform plane wave propagating within the cavity satisfying the following relation
\begin{align*}
\frac{2\omega n L}{c} &= 2m\pi
\end{align*}
where ω is the angular frequency, n is the effective refractive index and L is the length of the laser cavity. I have derived the group refractive index as follows
\begin{align*}
n_g &= n + \omega \frac{dn}{d\omega} \\
&= \frac{c m \pi}{\omega L} - \frac{c m \pi}{ \omega L}\\
&= 0
\end{align*}

If this is correct, I don't understand what this is physically entailing. Any insight would be much appreciated! Thank you.

2. May 11, 2016

### blue_leaf77

You are calculating the group index of refraction for a monochromatic wave, which is of course zero.